Hyperbolic Functions
Hyperbolic cosine is $$${\color{red}{{{y}={\cosh{{\left({x}\right)}}}=\frac{{{{e}}^{{x}}+{{e}}^{{-{x}}}}}{{2}}}}}$$$.
Hyperbolic sine is $$${\color{blue}{{{y}={\sinh{{\left({x}\right)}}}=\frac{{{{e}}^{{x}}-{{e}}^{{-{x}}}}}{{2}}}}}$$$.
Hyperbolic tangent is $$${\color{green}{{{y}={\tanh{{\left({x}\right)}}}=\frac{{{\sinh{{\left({x}\right)}}}}}{{{\cosh{{\left({x}\right)}}}}}=\frac{{{{e}}^{{x}}-{{e}}^{{-{x}}}}}{{{{e}}^{{x}}+{{e}}^{{-{x}}}}}}}}$$$.
Hyperbolic cotangent is $$$y=\coth\left(x\right)=\frac{\cosh\left(x\right)}{\sinh\left(x\right)}=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$$$.
Hyperbolic secant is $$${y}=\operatorname{sech}{\left({x}\right)}=\frac{{1}}{{{\cosh{{\left({x}\right)}}}}}=\frac{{2}}{{{{e}}^{{x}}+{{e}}^{{-{x}}}}}$$$ .
Hyperbolic cosecant is $$${y}={\operatorname{csch}{{\left({x}\right)}}}=\frac{{1}}{{{\sinh{{\left({x}\right)}}}}}=\frac{{2}}{{{{e}}^{{x}}-{{e}}^{{-{x}}}}}$$$.
There is some similarity between hyperbolic functions and trigonometric.
Domain of hyperbolic functions is $$${\left(-\infty,\infty\right)}$$$, except for function $$${y}={\coth{{\left({x}\right)}}}$$$ which is undefined when $$${x}={0}$$$.
Formulas that hold for any $$${x}$$$ and $$${y}$$$:
- $$${\cosh{{\left({x}\pm{y}\right)}}}={\cosh{{\left({x}\right)}}}{\cosh{{\left({y}\right)}}}\pm{\sinh{{\left({x}\right)}}}{\sinh{{\left({y}\right)}}}$$$.
- $$${\sinh{{\left({x}\pm{y}\right)}}}={\sinh{{\left({x}\right)}}}{\cosh{{\left({y}\right)}}}\pm{\cosh{{\left({x}\right)}}}{\sinh{{\left({y}\right)}}}$$$.
- $$${{\cosh}}^{{2}}{\left({x}\right)}-{{\sinh}}^{{2}}{\left({x}\right)}={1}$$$.
- $$${\cosh{{\left({2}{x}\right)}}}={{\cosh}}^{{2}}{\left({x}\right)}+{{\sinh}}^{{2}}{\left({x}\right)}$$$.
- $$${\sinh{{\left({2}{x}\right)}}}={2}{\sinh{{\left({x}\right)}}}{\cosh{{\left({x}\right)}}}$$$.
This formulas can be easily proved using definitions of hyperbolic functions.